As we are dealing with three-manifolds, it is natural to approach this question from the point of view of Thurston's geometrization. Using tools from involutive Heegaard Floer homology, it was shown in [1]} that Seifert fibered spaces do not generate \(\Theta _3\) , and in fact the quotient of \(\Theta _3\)  by Seifert spaces is infinitely generated [2]}. In particular, Seifert manifolds are too narrow of a class to understand the general properties of \(\Theta _3\) .
